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Completely disagree. Math isn't manipulation of symbols. At all. Math is the study of mathematical objects, a practice often done using a formal language for th

by tel 16y ago

Completely disagree. Math isn't manipulation of symbols. At all. Math is the study of mathematical objects, a practice often done using a formal language for the convenience and power it provides. "0" is a formal symbol with particular formal behavior "empty/missing/none" is a well-known physical concept Zero is a precise, powerful mathematical object which can be represented by them both. --- This is difficult to deny. Unless you want to deny the providence of most widely recognized mathematicians throughout history, you have to accept that formal language of math is relatively new. Furthermore, it's alive and growing, inconsistant and incomplete. There is a meaningful frontier, and there you observe mathematicians are really studying something else and furiously creating the formal language to describe it. In this light, metaphor is absolutely a useful tool in the same class as formal language for explaining and reasoning about math. You're right to point out the non-equivalence of the two, but the author's Kill Math project is in no way not math. Furthermore, I'm anecdotally a supporter of the author's belief that doing math competently requires knowing a the metaphorical side since your symbolic projects may fail or be unclear. I'd be willing to accept that metaphor will never be as powerful as formal language, but it does discredit to the way (I'd wager) most people understand math to deny the metaphorical. --- At the heart of this trouble of definitions is Gödel's Incompletenesses. The practical effect of their discovery was the destruction of the dreams of formalists who had for years hope to discover the essential shape of the formal language from which all math would spring. With Incompleteness however, we are forced to admit that we can study, meaningfully, the behavior of mathematical objects for which the language of math cannot be used to reason about. Then we extend that language, of course.

mechanical_fish 16y ago
I like this response. I have no time to craft a nuanced reply so let me just take a shortcut and concede: If you can get the student from "here are a bunch of physical concepts" to "here is a mathematical object, with interesting abstract properties that you can reason about" without introducing formalism you'll have succeeded in teaching math. Excellent. Is this plan really going to work very often? It is easy to say that you can derive, say, the utility of "zero" without ever doing any arithmetic -- just as it is easy to say that you can be a full-fledged computer scientist without ever touching a computer -- but in practice? It's true that the presence of a supercomputer in everyone's pocket will change this argument significantly. But simulations go only so far. They, too, are only metaphors, and if you don't know enough to tinker under their covers they are rather inflexible metaphors. Your classical mechanics simulator is not going to discover quantum mechanics for you. In my experience learning about mathematical abstractions requires all of the above tools -- you tinker with the formalism, you ponder the physical analogies, you draw mental pictures of clouds and colors, you play with a simulator, you build some circuits in the lab, you go for a walk, you tinker with the formalism again, and six years later you finally get it.
- tel 16y ago
  And I like this one! I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In particular, it's easy to see how formal descriptions can push mathematical generalization forward far before we have a suitable concept of the mathematical object we're describing. I think we're all (incl. the op) in some kind of agreement here about the didacticism of math. The op didn't disregard the power and utility of mathematical languages — he came from being trained pretty heavily in engineering math, at least up to playing with higher-order differential equations — but instead was, perhaps not directly, arguing for increased metaphorical/physical descriptions in taught mathematics. He's just responding to the rather eye-opening feeling one gets when one starts to realize that math is so interpretable! I think that's a perfectly fair argument to have. I know that in my own experience, I never understood the joy of math until the day that linear algebra took an interpretation as linear space transformation. --- So we're just sort of all oscillating here in strong rebuttals of whatever interpretation of the "heart" of math the prior author champions for a while. Which is fun but unproductive. --- I like the challenge of taking someone from physical concepts and metaphors directly to a mathematical object. I think it'd be possible, and maybe even useful when someone first starts to learn real math, but certainly it's not the most efficient way to become well-read. It'd be a lot like explaining the meaning of, I dunno, Día de los Muertos without immersing someone in Mexican language and culture. A single point of contact can be forced, but you lose so much context and fluency.
  - scott_s 16y ago
    I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.
3am 16y ago
Are you sure you're accurately describing Godel's Incompleteness theorem? It's been a while, but I understood it at a technical level briefly. I'd be interested in knowing more about how you think it applies here.
- jongraehl 16y ago
  He's not.
- tel 16y ago
  No, I'm not. I'm interpreting it pretty loosely. Sorry about that. Correct me if I'm wrong here, though. More directly, what I meant to say is that since there exist true theorems which cannot be proven within any particular choice of mathematical formalism, we need to operate with tools beyond simply symbolic manipulation. That was the death knell of Hilbert's Program and solidly separated the formal specification of math from "that thing which we're studying".
  - 3am 16y ago
    I think you are wrong - even though there are true, unprovable statements, they're unusual and to the best of my knowledge, not the subject of extensive study. More like curiosities (http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC http://en.wikipedia.org/wiki/List_of_statements_undecidable_...). Even if that were not the case, there is no alternate approach to studying them. I'm not really sure that you mean about the "formal specification" for math vs. the "that thing we're studying". An informal (ie, not expressed in ZFC + 1rst order predicate calculus) proof of something non-trivial can go on for dozens, if not hundreds of dense pages of symbols. If I recall correctly, Whitehead's Principae Mathematica derived arithmetic from ZCF and predicate calculus, and it took the whole book. I did a little reading to refresh myself on the subject, and this stood out as a good summary of the topic: "In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided." (http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foun...)

by tel 16y ago

mechanical_fish 16y ago

I like this response. I have no time to craft a nuanced reply so let me just take a shortcut and concede: If you can get the student from "here are a bunch of physical concepts" to "here is a mathematical object, with interesting abstract properties that you can reason about" without introducing formalism you'll have succeeded in teaching math. Excellent. Is this plan really going to work very often? It is easy to say that you can derive, say, the utility of "zero" without ever doing any arithmetic -- just as it is easy to say that you can be a full-fledged computer scientist without ever touching a computer -- but in practice? It's true that the presence of a supercomputer in everyone's pocket will change this argument significantly. But simulations go only so far. They, too, are only metaphors, and if you don't know enough to tinker under their covers they are rather inflexible metaphors. Your classical mechanics simulator is not going to discover quantum mechanics for you. In my experience learning about mathematical abstractions requires all of the above tools -- you tinker with the formalism, you ponder the physical analogies, you draw mental pictures of clouds and colors, you play with a simulator, you build some circuits in the lab, you go for a walk, you tinker with the formalism again, and six years later you finally get it.

tel 16y ago
And I like this one! I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In particular, it's easy to see how formal descriptions can push mathematical generalization forward far before we have a suitable concept of the mathematical object we're describing. I think we're all (incl. the op) in some kind of agreement here about the didacticism of math. The op didn't disregard the power and utility of mathematical languages — he came from being trained pretty heavily in engineering math, at least up to playing with higher-order differential equations — but instead was, perhaps not directly, arguing for increased metaphorical/physical descriptions in taught mathematics. He's just responding to the rather eye-opening feeling one gets when one starts to realize that math is so interpretable! I think that's a perfectly fair argument to have. I know that in my own experience, I never understood the joy of math until the day that linear algebra took an interpretation as linear space transformation. --- So we're just sort of all oscillating here in strong rebuttals of whatever interpretation of the "heart" of math the prior author champions for a while. Which is fun but unproductive. --- I like the challenge of taking someone from physical concepts and metaphors directly to a mathematical object. I think it'd be possible, and maybe even useful when someone first starts to learn real math, but certainly it's not the most efficient way to become well-read. It'd be a lot like explaining the meaning of, I dunno, Día de los Muertos without immersing someone in Mexican language and culture. A single point of contact can be forced, but you lose so much context and fluency.
- scott_s 16y ago
  I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.

3am 16y ago

Are you sure you're accurately describing Godel's Incompleteness theorem? It's been a while, but I understood it at a technical level briefly. I'd be interested in knowing more about how you think it applies here.

jongraehl 16y ago
He's not.
tel 16y ago
No, I'm not. I'm interpreting it pretty loosely. Sorry about that. Correct me if I'm wrong here, though. More directly, what I meant to say is that since there exist true theorems which cannot be proven within any particular choice of mathematical formalism, we need to operate with tools beyond simply symbolic manipulation. That was the death knell of Hilbert's Program and solidly separated the formal specification of math from "that thing which we're studying".
- 3am 16y ago
  I think you are wrong - even though there are true, unprovable statements, they're unusual and to the best of my knowledge, not the subject of extensive study. More like curiosities (http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC http://en.wikipedia.org/wiki/List_of_statements_undecidable_...). Even if that were not the case, there is no alternate approach to studying them. I'm not really sure that you mean about the "formal specification" for math vs. the "that thing we're studying". An informal (ie, not expressed in ZFC + 1rst order predicate calculus) proof of something non-trivial can go on for dozens, if not hundreds of dense pages of symbols. If I recall correctly, Whitehead's Principae Mathematica derived arithmetic from ZCF and predicate calculus, and it took the whole book. I did a little reading to refresh myself on the subject, and this stood out as a good summary of the topic: "In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided." (http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foun...)